On the fast Khintchine spectrum in continued fractions

TL;DR

This paper determines the Hausdorff dimension of a set of numbers in [0,1) characterized by a specific growth rate of the sum of logarithms of their continued fraction partial quotients, with no extra conditions on the growth function.

Contribution

It provides a complete characterization of the fast Khintchine spectrum for arbitrary functions growing faster than linearly, without additional restrictions.

Findings

Hausdorff dimension of E(ψ) is explicitly determined

The result holds for any ψ with ψ(n)/n→∞

No extra conditions on ψ are required

Abstract

For $x\in [0,1)$, let $x=[a_1(x), a_2(x),...]$ be its continued fraction expansion with partial quotients ${a_n(x), n\ge 1}$. Let $\psi : \mathbb{N} \rightarrow \mathbb{N}$ be a function with $\psi(n)/n\to \infty$ as $n\to \infty$. In this note, the fast Khintchine spectrum, i.e., the Hausdorff dimension of the set $$ E(\psi):=\Big{x\in [0,1): \lim_{n\to\infty}\frac{1}{\psi(n)}\sum_{j=1}^n\log a_j(x)=1\Big} $$ is completely determined without any extra condition on $\psi$.